SCIENTIA MAGNA, book series, Vol. 5, No. 1

8/9/2019 SCIENTIA MAGNA, book series, Vol. 5, No. 1

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Vol. 5 Some properties of ( α, β )-fuzzy B G -algebras 117

(i) ( , q ), (ii) ( , q ),

(iii) ( q, ), (iv) ( q, q ),

(v) ( q, q ), (vi) ( q, q ),

(vii) ( q, ).

Proof. Let µ = χ S . We show that µ is ( , q )-fuzzy subalgebra. Let xt 1 µ andx t 2 µ, for t1 , t 2 (0, 1]. Then µ(x) ≥ t1 and µ(y) ≥ t2 imply that x, y S . Thusx y S , i.e µ(x y) = 1. Therefore µ(x y) ≥ min( t1 , t 2) and µ(x y) + min( t 1 , t 2) > 1,i.e (x y)min( t 1 ,t 2 ) qµ. Similar to above argument, we can see that µ is an (α, β )-fuzzysubalgebra of X , where (α, β ) is one of the above forms.Conversely, we show that µ = χ X 0 . Suppose that there exists x X 0 such that µ(x) < 1. Letα = , choose t (0, 1] such that t < min(1 − µ(x), µ (x), µ (0)). Then xt αµ and 0 t αµ , but(x 0)min( t,t ) = xt βµ , where β = q or β = q . Which is a contradiction. If α = q , then

x1αµ and 01αµ , while (x 0)min(1 ,1) = x1βµ where β = or β = q , which is a contradiction.Now let α = q and choose t (0, 1] such that xt µ but x t qµ. Then xt αµ and 01αµ but(x 0)min( t, 1) = x t βµ for β = q or β = q , which is a contradiction. Finally we have x 1 qµand 01 qµ but ( x 0)min(1 ,1) = x1 µ, which is a contradiction. Therefore µ = χ X 0 .

Theorem 3.16. Let S be a subalgebra of X and let µ be a fuzzy set of X such that

(a) µ(x ) = 0 for all x X \ S ,

(b) µ(x) ≥ 0.5 for all x S .

Then µ is a (q, q )-fuzzy subalgebra of X .

Proof. Let x, y X and t1 , t 2 (0, 1] be such that xt 1 qµ and yt 2 qµ. Then we get thatµ(x) + t

1 > 1 and µ(y) + t

2 > 1. We can conclude that x y S , since in otherwise x X \ S

or y X \ S and therefore t1 > 1 or t2 > 1 which is a contradiction. If min( t 1 , t 2) > 0.5, thenµ(x y)+min( t1 , t 2) > 1 and so (x y)min( t 1 ,t 2 ) qµ. If min( t 1 , t 2) ≤ 0.5, then µ(x y) ≥ min( t1 , t 2)and thus ( x y)min( t 1 ,t 2 ) µ. Hence (x y)min( t 1 ,t 2 ) qµ.

Theorem 3.17. Let µ be a (q, q )-fuzzy subalgebra of X such that µ is not constanton the set X 0 . Then there exists x X such that µ(x) ≥ 0.5. Moreover, µ(x) ≥ 0.5 for allx X 0 .

Proof. Assume that µ(x) < 0.5 for all x X . Since µ is not constant on X 0 , thenthere exists x X 0 such that tx = µ(x) = µ(0) = t0 . Let t0 < t x . Choose δ > 0.5 suchthat t0 + δ < 1 < t x + δ . It follows that xδ qµ, µ(x x) = µ(0) = t0 < δ = min( δ, δ ) andµ(x x)+min( δ, δ ) = µ(0)+ δ = t

0+ δ < 1. Thus ( x x)

min( δ,δ ) qµ, which is a contradiction.

Now, if tx < t 0 then we can choose δ > 0.5 such that tx + δ < 1 < t 0 + δ . Thus 0 δ qµ andx1qµ, but ( x 0)min(1 ,δ ) = x δ qµ, because µ(x) < 0.5 < δ and µ(x) + δ = t x + δ < 1, whichis a contradiction. Hence µ(x) ≥ 0.5 for some x X . Now we show that µ(0) ≥ 0.5. On thecontrary, assume that µ(0) = t 0 < 0.5. Since there exists x X such that µ(x) = t x ≥ 0.5, itfollows that t0 < t x . Choose t1 > t 0 such that t0 + t1 < 1 < t x + t1 . Then µ(x)+ t1 = tx + t1 > 1,and so x t qµ. Thus we can conclude that

µ(x x) + min( t1 , t 1) = µ(0) + t1 = t0 + t1 < 1,






t > 0 such that µ(x y) < t < min(µ(x), µ(y)). Thus xt µ and yt µ but ( x y)min( t,t ) =(x y)t qµ, since µ(x y) < t and µ(x y)+ t < 1 < 2t < 1, which is a contradiction. Hence if min( µ(x), µ (y)) < 0.5, then µ(x y) ≥ min(µ(x), µ (y)). If min( µ(x), µ (y)) ≥ 0.5, then x0.5 µ

and y0.5 µ. So we can get that

(x y)min(0 .5,0.5) = ( x y)0.5 qµ.

Then µ(x y) > 0.5. Consequently, µ(x y) ≥ min(µ(x), µ(y), 0.5), for all x, y X .Conversely, let x, y X and t1 , t 2 (0, 1] be such that xt 1 µ and yt 2 µ. So µ(x ) ≥ t1

and µ(y) ≥ t2 . Then by hypothesis we have µ(x y) ≥ min(µ(x ), µ (y), 0.5) ≥ min( t1 , t 2 , 0.5).If min( t1 , t 2) ≤ 0.5, then µ(x y) ≥ min(µ(x), µ (y)). If min( t1 , t 2) > 0.5, then µ(x y) ≥ 0.5.Thus µ(x y) + min( t1 , t 2) > 1. Therefore ( x y)min( t 1 ,t 2 ) qµ.

Theorem 3.22. Let µ be an ( , q )-fuzzy subalgebra of X .(i) If there exists x X such that µ(x ) ≥ 0.5, then µ(0) ≥ 0.5;(ii) If µ(0) < 0.5, then µ is an ( , )-fuzzy subalgebra of X .Proof. (i) Let µ(x) ≥ 0.5. Then by hypothesis we have µ(0) = µ(x x ) ≥ min(µ(x), µ(x ), 0.5) =

0.5.(ii) Let µ(0) < 0.5. Then by (i) µ(x) < 0.5, for all x X . Now let xt 1 µ and yt 2 µ,

for t1 , t 2 (0, 1]. Then µ(x) ≥ t1 and µ(y) ≥ t2 . Thus µ(x y) ≥ min(µ(x), µ (y), 0.5) ≥min( t1 , t 2 , 0.5) = min( t1 , t 2). Therefore ( x y)min( t 1 ,t 2 ) µ.

Lemma 3.23. Let µ be a non-zero ( , q ) fuzzy subalgebra of X . Let x, y X suchthat µ(x ) < µ (y). Then

µ(x y) =µ(x ) if µ(y) < 0.5 or µ (x) < 0.5 ≤ µ(y)

≥ 0.5 if µ(x) ≥ 0.5.

Proof. Let µ(y) < 0.5. Then we have µ(x y) ≥ min(µ(x ), µ (y), 0.5) = µ(x). Alsoµ(x) = µ((x y) (0 y)) ≥ minµ(x y), µ(0 y), 0.5 (1)

Now we show that µ(0 y) ≥ µ(y). Since µ(y) < 0.5, then µ(0) = µ(y y) ≥ minµ(y), µ(y), 0.5 =µ(y). Thus µ(0 y) ≥ minµ(0) , µ(y), 0.5 = µ(y). Hence (1) and hypothesis imply thatµ(x) ≥ minµ(x y), µ (y). Since µ(x) < µ (y), then µ(x) ≥ µ(x y). Therefore µ(x y) = µ(x).Now let µ(x ) < 0.5 ≤ µ(y). Then similar to above argument µ(x y) ≥ µ(x) and µ(x) ≥minµ(x y), µ(0 y), 0.5. Since µ(y) ≥ 0.5, then by Theorem 3.22(i), µ(0) ≥ 0.5. Thusµ(0 y) ≥ minµ(0) , µ (y), 0.5 = 0 .5. So by hypothesis we get that µ(x) ≥ minµ(x y), 0.5.Thus µ(x) < 0.5 imply that µ(x) ≥ µ(x y). Therefore µ(x y) = µ(x). Let µ(x ) ≥ 0.5. Thenµ(x y) ≥ min(µ(x), µ(y), 0.5) = 0 .5.

Theorem 3.24. Let µ be an ( , q )-fuzzy subalgebra of X . Then for all t [0, 0.5],the nonempty level set U (µ; t) is a subalgebra of X . Conversely, if the nonempty level set µ isa subalgebra of X , for all t [0, 1], then µ is an ( , q )-fuzzy subalgebra of X .

Proof. Let µ be an ( , q )-fuzzy subalgebra of X . If t = 0, then U (µ; t) is a subalgebraof X . Now let U (µ; t ) = , 0 < t ≤ 0.5 and x, y U (µ; t). Then µ(x), µ(y) ≥ t. Thus byhypothesis we have µ(x y) ≥ min(µ(x ), µ (y), 0.5) ≥ min( t, 0.5) ≥ t. Therefore U (µ; t) is asubalgebra of X .



120 L. Torkzadeh and A. Borumand Saeid No. 1

Conversely, let x, y X . Then we have

µ(x), µ (y) ≥ min(µ(x), µ(y), 0.5) = t0 .

Hence x, y U (µ; t0), for t0 [0, 1] and so x y U (µ; t0). Therefore µ(x y) ≥ t0 =min( µ(x), µ (y), 0.5), i.e µ is an ( , q )-fuzzy subalgebra of X .

Theorem 3.25. Let S be a subset of X . The characteristic function χ S of S is an( , q )-fuzzy subalgebra of X if and only if S is a subalgebra of X .

Proof. Let X S be an ( , q )-fuzzy subalgebra of X and x, y S . Then χ S (x) = 1 =χ S (y), and so x1 χ S and y1 χ S . Hence (x y)1 = ( x y)min(1 ,1) qχ S , which implies thatχ S (x y) > 0. Thus x y S . Therefore S is a subalgebra of X .

Conversely, if S is a subalgebra of X , then χ S is an ( , )-fuzzy subalgebra of X . So byTheorem 3.5 we get that µ is an ( , q )-fuzzy subalgebra of X .

Lemma 3.26. Let f : X → Y be a BG -homomorphism and G be a fuzzy set of Y with

membership function µG . Then xt αµ f − 1 (G ) f (x) t αµ G , for all α , q, q, q .Proof. Let α = . Then

x t αµ f − 1 (G ) µf − 1 (G ) (x) ≥ t µG (f (x)) ≥ t (f (x)) t αµ G .

The proof of the other cases is similar to above argument.Theorem 3.27. Let f : X → Y be a BG -homomorphism and G be a fuzzy set of Y with

membership function µG .(i) If G is an (α, β )-fuzzy subalgebra of Y , then f − 1(G ) is an (α, β )-fuzzy subalgebra of X ,(ii) Let f be epimorphism. If f − 1(G ) is an (α, β )-fuzzy subalgebra of X , then G is an

(α, β )-fuzzy subalgebra of Y .

Proof. (i) Let xt αµ f − 1

(G ) and yr αµ f − 1

(G ) , for t, r (0, 1]. Then by Lemma 3.26, weget that ( f (x)) t αµ G and ( f (y)) r αµ G . Hence by hypothesis ( f (x) f (y))min( t,r ) βµ G . Then(f (x y)) min( t,r ) βµ G and so (x y)min( t,r ) βµ f − 1 (G ) .

(ii) Let x, y Y . Then by hypothesis there exist x , y X such that f (x ) = x and f (y ) =y. Assume that x t αµ G and yr αµ G , then ( f (x )) t αµ G and ( f (y )) r αµ G . Thus xt αµ f − 1 (G ) andyr αµ f − 1 (G ) and therefore ( x y )min( t,r ) βµ f − 1 (G ) . So

(f (x y ))min( t,r ) βµ G (f (x ) f (y )) min( t,r ) βµ G (x y)min( t,r ) βµ G .

Theorem 3.28. Let f : X → Y be a BG -homomorphism and H be an ( , q )-fuzzysubalgebra of X with membership function µH . If µH is an f -invariant, then f (H ) is an

( , q )-fuzzy subalgebra of Y .Proof. Let y1 and y2 Y . If f − 1(y1) or f − 1(y2) = , then µf (H ) (y1 y2) ≥ min(µf (H ) (y1), µf (H ) (y2), 0.5).Now let f − 1(y1) and f − 1(y2) = . Then there exist x1 , x2 X such that f (x1) = y1 andf (x 2) = y2 . Thus by hypothesis we have

µf (H ) (y1 y2) = supt f − 1 (y 1 y 2 )

µH (t)

= supt f − 1 ( f (x 1 x 2 ))

µH (t )




= µH (x1 x2) since µH is an f -invariant

≥ min(µH (x1), µH (x2), 0.5)

= min( supt f − 1 (y 1 )

µH (t), supt f − 1 (y 2 )

µH (t), 0.5)

= min( µf (H ) (y1), µf (H ) (y2), 0.5).

So by Theorem 3.21, f (H ) is an ( , q )-fuzzy subalgebra of Y .Lemma 3.29. Let f : X → Y be a BG -homomorphism.(i) If S is a subalgebra of X , then f (S ) is a subalgebra of Y ;(ii) If S is a subalgebra of Y , then f − 1(S ) is a subalgebra of X .Proof. The proof is easy.

Theorem 3.30. Let f : X → Y be a BG -homomorphism . If H is a non-zero (q, q )-fuzzysubalgebra of X with membership function µH , then f (H ) is a non-zero ( q, q )-fuzzy subalgebraof Y .

Proof. Let H be a non-zero (q, q )-fuzzy subalgebra of X . Then by Theorem 3.10, we have

µH (x) =µH (0) if x X 0

0 otherwise. Now we show that µ f (H ) (y) =

µH (0) if y f (X 0)

0 otherwise.

Let y Y . If y f (X 0), then there exist x X 0 such that f (x) = y. Thus µf (H ) (y) =sup

t f − 1 (y )µH (t) = µH (0). If y f (X 0), then it is clear that µf (H ) (y) = 0. Since X 0 is subal-

gebra of X , then f (X 0) is a subalgebra of Y . Therefore by Theorem 3.11, f (H ) is a non-zero(q, q )-fuzzy subalgebra of Y .

Theorem 3.31. Let f : X → Y be a BG -homomorphism . If H is an (α, β )-fuzzy subal-gebra of X with membership function µH , then f (H ) is an (α, β )-fuzzy subalgebra of Y , where(α, β ) is one of the following form

(i) ( , q ), (ii) ( , q ),

(iii) ( q, ), (iv) ( q, q ),

(v) ( q, q ), (vi) ( q, q ),

(vii) ( q, ), (viii) ( q, q ).

Proof. The proof is similar to the proof of Theorem 3.30, by using of Theorems 3.15 and

3.18.Theorem 3.32. Let f : X → Y be a BG -homomorphism and H be an ( , )-fuzzysubalgebra of X with membership function µH . If µH is an f -invariant, then f (H ) is an( , )-fuzzy subalgebra of Y .

Proof. Let zt µf (H ) and yr µf (H ) , where t, r (0, 1]. Then µf (H ) (z) ≥ t andµf (H ) (y) ≥ r . Thus f − 1(z), f − 1(y) = imply that there exists x 1 , x 2 X such that f (x1) = zand f (x2) = y. since µH is an f -invariant, then µf (H ) (z) ≥ t and µf (H ) (y) ≥ r imply thatµH (x1) ≥ t and µH (x2) ≥ r . So by hypothesis we have



122 L. Torkzadeh and A. Borumand Saeid No. 1

µf (H ) (z y) = supt f − 1 (z y )

µH (t )

= supt f − 1 ( f (x 1 x 2 ))

µH (t)

= µH (x1 x2)

≥ min( t, r ).

Therefore ( z y)min( t,r ) µf (H ) , i.e f (H ) is an ( , )-fuzzy subalgebra of Y .Theorem 3.33. Let µ i | i Λ be a family of ( , q )-fuzzy subalgebra of X . Then

µ :=i Λ

µi is an ( , q )-fuzzy subalgebra of X .

Proof. By Theorem 3.21 we have, for all i Λ

µi (x y) ≥ min(µi (x), µ i (y), 0.5).

Therefore µ(x y) = inf i Λ

µ i (x y) ≥ inf i Λ

min( µ i (x), µ i (y), 0.5)

= min(inf i Λ

µi (x), inf i Λ

µi (y), 0.5)

= min( µ(x), µ(y), 0.5).

Therefore by Theorem 3.21, µ is an ( , q )-fuzzy subalgebra.Theorem 3.34. Let µi | i Λ be a family of ( , )-fuzzy subalgebra of X . Then

µ := i Λµi is an ( , )-fuzzy subalgebra of X .

Proof. Let xt µ and yr µ, t, r (0, 1]. Then µ(x) ≥ t and µ(y) ≥ r . Thus for alli Λ, µi (x) ≥ t and µ i (y) ≥ r imply that µi (x y) ≥ min( t, r ). Therefore µ(x y) ≥ min( t, r )i.e (x y)min( t,r ) µ.

Theorem 3.35. Let µi | i Λ be a family of (α, β )-fuzzy subalgebra of X . Thenµ :=

i Λ

µi is an (α, β )-fuzzy subalgebra of X , where (α, β ) is one of the following form

(i) ( , q ), (ii) ( , q ),

(iii) ( q, ), (iv) ( q, q ),

(v) ( q, q ), (vi) ( q, q ),(vii) ( q, ), (viii) ( q, q ),

(ix) ( q, q ).Proof. We prove theorem for ( q, q )-fuzzy subalgebra. The proof of the other cases is

similar, by using Theorems 3.15 and 3.18.If there exists i Λ such that µi = 0, then µ = 0. So µ is a (q, q )-fuzzy subalgebra. Let

µi = 0 for all i Λ. Then by Theorem 3.10 we have µi (x) =µ i (0) if x X i00 otherwise

, for all




i Λ. So it is clear that µ(x) =µ(0) if x

i Λ

X i0

0 otherwise. Since

i Λ

X i0 is a subalgebra of X ,

then by Theorem 3.11 µ is a (q, q )-fuzzy subalgebra of X .

References

[1] S. S. Ahn and H. D. Lee, Fuzzy Subalgebras of BG -algebras, Commun. Korean Math.Soc., 19 (2004) 243-251.

[2] S. K Bhakat and P. Das, ( , q )-fuzzy subgroups, Fuzzy Sets and Systems, 80 (1996),359-368.

[3] Y. Imai and K. Iseki, On axiom systems of propositional calculi, XIV Proc. JapanAcademy, 42 (1966), 19-22.

[4] C. B. Kim, H. S. Kim, On BG -algebras, (submitted).[5] J. Meng and Y. B. Jun, BCK-algebras, Kyung Moonsa, Seoul, Korea, 1994.[6] J. Neggers and H. S. Kim, On B -algebras, Math. Vensik, 54 (2002), 21-29.[7] J. Neggers and H. S. Kim, On d-algebras, Math. Slovaca, 49 (1999), 19-26.[8] P. M. Pu and Y. M. Liu, Fuzzy Topology I, Neighborhood structure of a fuzzy point

and Moore-Smith convergence, J. Math. Anal. Appl., 76 (1980), 571-599.[9] A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl., 35 (1971), 512-517.[10] L. A. Zadeh, Fuzzy Sets, Inform. Control, 8(1965), 338-353.



Scientia Magna

Vol. 5 (2009), No. 1, 124-127

On the Smarandache totient functionand the Smarandache power sequence

Yanting Yang and Min Fang

Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China

Abstract For any positive integer n , let SP (n ) denotes the Smarandache power sequence.And for any Smarandache sequence a(n ), the Smarandache totient function St (n ) is denedas (a (n )), where (n ) is the Euler totient function. The main purpose of this paper is

using the elementary and analytic method to study the convergence of the function S 1S 2

, where

S 1 =n

k =1

1St (k)

2

, S 2 = n

k =1

1St (k )

2

, and give an interesting limit Theorem.

Keywords Smarandache power function, Smarandache totient function, convergence.

§1. Introduction and results

For any positive integer n , the Smarandache power function SP (n ) is dened as the smallestpositive integer m such that n | mm , where m and n have the same prime divisors. That is,

SP (n ) = min m : n | mm , m N, p | n

p = p | m

p .

For example, the rst few values of SP (n ) are: SP (1) = 1, SP (2) = 2, SP (3) = 3, SP (4) = 2,SP (5) = 5, SP (6) = 6, SP (7) = 7, SP (8) = 4, SP (9) = 3, SP (10) = 10, SP (11) = 11,SP (12) = 6, SP (13) = 13, SP (14) = 14, SP (15) = 15, · · · . In reference [1], ProfessorF.Smarandache asked us to study the properties of SP (n ). It is clear that SP (n ) is not amultiplicative function. For example, SP (8) = 4 , SP (3) = 3 , SP (24) = 6 = SP (3) × SP (8).But for most n , we have SP (n ) =

p | n

p, where p | n

denotes the product over all different prime

divisors of n. If n = pα , k · pk + 1 ≤ α ≤ (k + 1) pk +1 , then we have SP (n ) = pk +1 , where0 ≤ k ≤ α − 1. Let n = pα 1

1 pα 2

2 · · · pα r

r , for all α i (i = 1 , 2, · · · , r), if α i ≤ pi , thenSP (n ) =

p | n

p.

About other properties of the function SP (n ), many authors had studied it, and gave someinteresting conclusions. For example, in reference [4], Zhefeng Xu had studied the mean valueproperties of SP (n ), and obtained a sharper asymptotic formula:

n ≤ x

SP (n ) = 12

x 2

p

1 − 1

p( p + 1)+ O x

32 + ,





126 Yanting Yang and Min Fang No. 1

By Lemma 1, we can easily get k

(k) = O(lnln k). Note that

k ≤ n

µ2 (k)k2 = O(1). And if

k B , then we can write k as k = l · m , where l is a square-free integer and m is a square-full

integer. Let S denotek B

1( (SP (k))) 2 , then from the properties of SP (k) and (k) we have

S ≤lm ≤ n

1

l2

p | m

p2

p | lm

1 − 1 p

2 =m ≤ n

1

p | m

p2l ≤ n

m

µ2 (l)l2 ·

l2 m 2

2 (lm ) = O (ln ln n )2

m ≤ n

1

p | m

p2.

Let U (k) = p | k

p, thenm ≤ n

1

p | m

p2=

k ≤ n

a (k)U 2 (k)

, where m is a square-full integer and the

arithmetical function a (k) is dened as follows:

a (k) =1, if k is a square-full integer;

0, otherwise.

Note that a(k)U 2 (k)

is a multiplicative function. According to the Euler product formula (see

reference [3] and [5]), we have

A(s) =∞

k =1

a (k)U 2 (k)k s =

p

1 + 1

p2+ s ( ps − 1).

From the Perron formulas [5], for b = 1 + 1ln n , T ≥ 1, we have

k ≤ n

a (k)U 2 (k)

= 12πi

b+ iT

b− iT A(s)

n s

s ds + O

n bζ (b)T

+ O n min 1, ln n

T +

a(n )2U 2 (n )

.

Taking T = n , we can get the estimate

On bζ (b)

T + O n min 1,

ln nT

+ a(n )2U 2 (n )

= O(ln n ).

Because the function A (s )n s

s is analytic in Re s > 0, taking c =

1ln n

, then we have

12πi

b+ iT

b− iT A(s)

n s

s ds +

b− iT

c− iT A(s)

n s

s ds +

c+ iT

b+ iT A(s )

n s

s ds +

c− iT

c+ iT A(s)

n s

s ds = 0 .

Note that c+ iT

c− iT A(s )

n s

s ds = O

T

− T

dy

c2 + y2 = O(ln n ) and

b− iT

c− iT A(s )

n s

s ds =

O b

c

n σ

T dσ = O

1ln n

. Similarly, b+ iT

c+ iT A(s)

n s

s ds = O

1ln n

. Hence,k ≤ n

a (k)U 2 (k)

=

O (ln n ).



Vol. 5 On the Smarandache totient function and the Smarandache power sequence 127

So

k ≤ n

1( (SP (k))) 2 = O(ln n · (ln ln n )2 ). (1)

Now we come to estimatek ≤ n

1(SP (k))

, from the denition of SP (n ), we may immediately

get that SP (n ) ≤ n . Let n = pα 1

1 pα 2

2 · · · pα s

s denotes the factorization of n into prime powers,then SP (n ) = pβ 1

1 pβ 2

2 · · · pβ s

s , where β i ≥ 1. Therefore, we can get that pβ 1 − 11 ( p1 − 1) pβ 2 − 1

2 ( p2 −1) · · · pβ s − 1

s ( ps − 1) ≤ pα 1 − 11 ( p1 − 1) pα 2 − 1

2 ( p2 − 1) · · · pα s − 1s ( ps − 1), thus ( pβ 1

1 pβ 2

2 · · · pβ s

s ) ≤( pα 1

1 pα 2

2 · · · pα s

s ). That is, (SP (n )) ≤ (n ), according to Lemma 2, we can easily get

k ≤ n

1(SP (k))

≥k ≤ n

1(k)

= ζ (2)ζ (3)

ζ (6) ln n + A + O

ln nn

. (2)

Combining (1) and (2), we obtain

0 ≤

n

k =1

1(SP (k))

2

n

k =1

1(SP (k))

2 ≤ O ln n · (ln ln n )2

ζ (2)ζ (3)ζ (6)

ln n + A + Oln n

n

2 −→ 0, as n → ∞ .

This completes the proof of our Theorem.

References

[1] F. Smarandache, Only Problems, Not Solutions, Xiquan Publishing House, Chicago,1993.

[2] F. Russo, A set of new Smarandache functions, sequences and conjectures in numbertheory, American Research Press, USA, 2000.

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[4] Huanqin Zhou, An innite series involving the Smarandache power function SP (n ),Scientia Magna, 2(2006), No.3, 109-112.

[5] Tom M. Apostol, Introduction to analytical number theory, Springer-Verlag, New York,1976.

[6] H. L. Montgomery, Primes in arithmetic progressions, Mich. Math. J., 17 (1970), 33-39.[7] Pan Chengdong and Pan Chengbiao, Foundation of analytic number theory, Science

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Scientia Magna

Vol. 5 (2009), No. 1, 128-132

A new additive function andthe F. Smarandache function

Yanchun Guo

Department of Mathematics, Xianyang Normal UniversityXianyang, Shaanxi, P.R.China

Abstract For any positive integer n , we dene the arithmetical function F (n ) as F (1) = 0.

If n > 1 and n = pα 1

1 pα 2

2 · · ·

p

α k

k be the prime power factorization of n , then F (n ) = α1

p1

+α 2 p2 + · · · + α k pk . Let S (n ) be the Smarandache function. The main purpose of this paperis using the elementary method and the prime distribution theory to study the mean valueproperties of ( F (n ) − S (n )) 2 , and give a sharper asymptotic formula for it.

Keywords Additive function, Smarandache function, Mean square value, Elementary method,Asymptotic formula.

§1. Introduction and result

Let f (n ) be an arithmetical function, we call f (n ) as an additive function, if for any positive

integers m , n with ( m, n ) = 1, we have f (mn ) = f (m ) + f (n ). We call f (n ) as a completeadditive function, if for any positive integers r and s, f (rs ) = f (r ) + f (s ). In elementarynumber theory, there are many arithmetical functions satisfying the additive properties. Forexample, if n = pα 1

1 pα 2

2 · · · pα kk denotes the prime power factorization of n , then function Ω( n ) =

α 1 + α 2 + · · ·+ α k and logarithmic function f (n ) = ln n are two complete additive functions,ω(n ) = k is an additive function, but not a complete additive function. About the propertiesof the additive functions, one can nd them in references [1], [2] and [5].

In this paper, we dene a new additive function F (n ) as follows: F (1) = 0; If n > 1 and n = pα 1

1 pα 2

2 · · · pα kk denotes the prime power factorization of n , then F (n ) = α 1 p1 + α 2 p2 + · · ·+ α k pk .

It is clear that this function is a complete additive function. In fact if m = pα 1

1 pα 2

2 · · · pα kk

and n

= pβ 1

1 pβ 2

2 · · · pβ k

k , then we have mn

= pα 1 + β 1

1 pα 2 + β 2

2 · · · pα k + β k

k . Therefore, F

(mn

) =(α 1 + β 1) p1 + ( α 2 + β 2) p2 + · · ·+ ( α k + β k ) pk = F (m ) + F (n ). So F (n ) is a complete additivefunction. Now we let S (n ) be the Smarandache function. That is, S (n ) denotes the smallestpositive integer m such that n divide m!, or S (n ) = min m : n | m!. About the propertiesof S (n ), many authors had studied it, and obtained a series results, see references [7], [8] and[9]. The main purpose of this paper is using the elementary method and the prime distributiontheory to study the mean value properties of ( F (n ) −S (n )) 2 , and give a sharper asymptoticformula for it. That is, we shall prove the following:

Theorem. Let N be any xed positive integer. Then for any real number x > 1, we



Vol. 5 A new additive function and the F. Smarandache function 129

have the asymptotic formula

n ≤x

(F (n )

−S (n )) 2 =

N

i =1

ci

·

x2

lni +1

x+ O

x2

lnN +2

√ x,

where ci (i = 1 , 2, · · · , N ) are computable constants, and c1 = π 2

6 .

§2. Proof of the theorem

In this section, we use the elementary method and the prime distribution theory to completethe proof of the theorem. We using the idea in reference [4]. First we dene four sets A, B ,C , D as follows: A = n, n N , n has only one prime divisor p such that p | n and p2 n , p > n

13 ; B = n, n N , n has only one prime divisor p such that p2 | n and p > n

13 ;

C = n, n N , n has two deferent prime divisors p1 and p2 such that p1 p2 | n , p2 > p 1 > n1

3

;D = n, n N , any prime divisor p of n satisfying p ≤ n

13 , where N denotes the set of all

positive integers. It is clear that from the denitions of A , B , C and D we have

n ≤x

(F (n ) −S (n )) 2 =n ≤xn A

(F (n ) −S (n ))2 +n ≤xn B

(F (n ) −S (n )) 2

+n ≤xn C

(F (n ) −S (n )) 2 +n ≤xn D

(F (n ) −S (n ))2

≡ W 1 + W 2 + W 3 + W 4 . (1)

Now we estimate W 1 , W 2 , W 3 and W 4 in (1) respectively. Note that F (n ) is a completeadditive function, and if n A with n = pk , then S (n ) = S ( p) = p, and any prime divisor q of k satisfying q ≤ n

13 , so F (k) ≤ n

13 ln n . From the Prime Theorem (See Chapter 3, Theorem 2

of [3]) we know that

π (x ) = p≤x

1 =k

i =1

ci · x

lni x+ O

x

lnk +1 x, (2)

where ci (i = 1 , 2, · · · , k) are computable constants, and c1 = 1. By these we have theestimate:

W 1 =n ≤xn A

(F (n ) −S (n )) 2 = pk ≤x

( pk ) A

(F ( pk) − p)2

= pk ≤x

( pk ) A

F 2(k) k≤√ x k<p ≤ x

k

( pk)23 ln2( pk) ≤ (ln x )2

k ≤√ xk

23

k<p ≤ xk

p23

(ln x )2

k ≤√ xk

23

xk

53 1

ln xk

x53 ln2 x. (3)



130 Yanchun Guo No. 1

If n B , then n = p2k , and note that S (n ) = S ( p2) = 2 p, we have the estimate

W 2 =n

≤x

n B

(F (n ) −S (n )) 2 = p 2 k

≤x

p>k

F ( p2k) −2 p2

=k≤x

13 k<p ≤√ xk

F 2(k) k ≤x

13 k<p ≤√ xk

k2

k≤x13

k2 ·x12

k12 ln x

x43

ln x. (4)

If n D , then F (n ) ≤ n13 ln n and S (n ) ≤ n

13 ln n , so we have

W 4 =n ≤xn D

(F (n ) −S (n )) 2

n ≤x

n23 ln2 n x

53 ln2 x. (5)

Finally, we estimate main term W 3 . Note that n C , n = p1 p2k , p2 > p 1 > n13 > k . If

k < p 1 < n13 , then in this case, the estimate is exact same as in the estimate of W 1 . If

k < p 1 < p 2 < n13 , in this case, the estimate is exact same as in the estimate of W 4 . So by (2)

we have

W 3 =n ≤xn C

(F (n ) −S (n )) 2 = p 1 p 2 k≤x p 2 >p 1 >k

(F ( p1 p2k) − p2)2 + O x53 ln2 x

=k ≤x

13 k<p 1 ≤√ xk p 2 ≤

xp 1 k

F 2(k) + 2 p1F (k) + p21 + O x

53 ln2 x

=k ≤x

13 k<p 1 ≤√ xk p 1 <p 2 ≤

xp 1 k

p21 + O

k ≤x13 k<p 1 ≤√ xk p 1 <p 2 ≤

xp 1 k

kp 1 + O x53 ln2 x

=k ≤x

13 k<p 1 ≤√ xk

p21

N

i =1

ci · x

p1k lni x p 1 k

+ O x

p1k lnN +1 x+ O x

53 ln2 x

−k≤x

13 k<p 1 ≤√ xk

p21

p 2 ≤ p 1

1 + O

k≤x13 k<p 1 ≤√ xk p 1 <p 2 ≤

xp 1 k

kp 1 . (6)

Note that ζ (2) = π 2

6 , from the Abel’s identity (See Theorem 4.2 of [6]) and (2) we have

k ≤x13 k<p 1 ≤√ xk

p21

p≤ p 1

1 =k ≤x

13 k<p 1 ≤√ xk

p21

N

i =1

ci · p1

lni p1+ O

p1

lnN +1 p1

=N

i =1 k≤x13 k<p 1 ≤√ xk

ci · p31

lni p1+ O

k ≤x13 k<p 1 ≤√ xk

p31

lnN +1 p1

=N

i =1

di ·x 2

lni +1 x+ O

2N ·x 2

lnN +2 x, (7)



Vol. 5 A new additive function and the F. Smarandache function 131

where d i (i = 1 , 2, · · · , N ) are computable constants, and d1 = π 2

6 .

k ≤x13 k<p 1 ≤√ xk p 1 <p 2 ≤

xp 1 k

kp 1 k ≤x

13

k

p 1 ≤√ xk p1 ·

x p1k ln x

k ≤x13

x32

√ k ln2 x

x53

ln2 x. (8)

k≤x13 k<p 1 ≤√ xk

p1x

k lnN +1 xk≤x

13

x 2

k2 lnN +2 x

x2

lnN +2 x. (9)

From the Abel’s identity and (2) we also have the estimate

k≤x13 k<p 1 ≤√ xk

p21

x p1k ln x

p 1 k=

k≤x13

1k

k<p 1 ≤√ xkxp 1

ln xkp 1

=N

i =1

bi

· x2

lni +1

x+ O

x2

lnN +1

x, (10)

where bi (i = 1 , 2, · · · , N ) are computable constants, and b1 = π 2

3 .Now combining (1), (3), (4), (5), (6), (7), (8)and(9) we may immediately deduce the

asymptotic formula:

n ≤x

(F (n ) −S (n ))2 =N

i =1

a i · x2

lni +1 x+ O

x2

lnN +2 √ x ,

where a i (i = 1 , 2, · · · , N ) are computable constants, and a 1 = b1 −d1 = π 2

6 .This completes the proof of Theorem.

References

[1] C.H.Zhong, A sum related to a class arithmetical functions, Utilitas Math., 44(1993),231-242.

[2] H.N.Shapiro, Introduction to the theory of numbers, John Wiley and Sons, 1983.[3] Pan Chengdong and Pan Chengbiao, The elementary proof of the prime theorem (in

Chinese), Shanghai Science and Technology Press, Shanghai, 1988.[4] Xu Zhefeng, On the value distribution of the Smarandache function, Acta Mathematica

Sinica (in Chinese), 49 (2006), No.5, 1009-1012.[5] Zhang Wenpeng, The elementary number theory (in Chinese), Shaanxi Normal Univer-

sity Press, Xi’an, 2007.[6] Tom M. Apostol. Introduction to Analytic Number Theory, Springer-Verlag, 1976.[7] Yi Yuan and Kang Xiaoyu, Research on Smarandache Problems (in Chinese), High

American Press, 2006.[8] Chen Guohui, New Progress On Smarandache Problems (in Chinese), High American

Press, 2007.[9] Liu Yanni, Li Ling and Liu Baoli, Smarandache Unsolved Problems and New Progress

(in Chinese), High American Press, 2008.



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